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Expand the expression to a polynomial in standard form:

left parenthesis, 2, x, squared, minus, x, plus, 3, right parenthesis, left parenthesis, x, squared, minus, 6, x, plus, 2, right parenthesis
(2x
2
−x+3)(x
2
−6x+2)

Expand the expression to a polynomial in standard form:
left parenthesis, x, squared, minus, x, minus, 2, right parenthesis, left parenthesis, minus, x, squared, plus, 3, x, minus, 6, right parenthesis
(x
2
−x−2)(−x
2
+3x−6)

minus, x, squared
−x
2
plus, 3, x
+3x
minus, 6
−6
x, squared
x
2
minus, x, to the power 4
−x
4
3, x, cubed
3x
3
minus, 6, x, squared
−6x
2
minus, x
−x
x, cubed
x
3
minus, 3, x, squared
−3x
2
6, x
6x
minus, 2
−2
2, x, squared
2x
2
minus, 6, x
−6x
12
12
Combine like terms:
Combine like terms:
minus, x, to the power 4 , plus, 4, x, cubed, minus, 7, x, squared, plus, 12
−x
4
+4x
3
−7x
2
+12

1 answer

Let's expand both expressions step by step to achieve the polynomial in standard form.

First Expression:

\[ (2x^2 - x + 3)(x^2 - 6x + 2) \]

  1. Distribute \(2x^2\): \[ 2x^2 \cdot (x^2 - 6x + 2) = 2x^4 - 12x^3 + 4x^2 \]

  2. Distribute \(-x\): \[ -x \cdot (x^2 - 6x + 2) = -x^3 + 6x^2 - 2x \]

  3. Distribute \(3\): \[ 3 \cdot (x^2 - 6x + 2) = 3x^2 - 18x + 6 \]

Now, combine all terms: \[ (2x^4 - 12x^3 + 4x^2) + (-x^3 + 6x^2 - 2x) + (3x^2 - 18x + 6) \]

Combine like terms:

  • \(2x^4\)
  • \((-12x^3 - x^3) = -13x^3\)
  • \((4x^2 + 6x^2 + 3x^2) = 13x^2\)
  • \((-2x - 18x) = -20x\)
  • Constant term: \(6\)

Final expression in standard form: \[ 2x^4 - 13x^3 + 13x^2 - 20x + 6 \]

Second Expression:

\[ (x^2 - x - 2)(-x^2 + 3x - 6) \]

  1. Distribute \(x^2\): \[ x^2 \cdot (-x^2 + 3x - 6) = -x^4 + 3x^3 - 6x^2 \]

  2. Distribute \(-x\): \[ -x \cdot (-x^2 + 3x - 6) = x^3 - 3x^2 + 6x \]

  3. Distribute \(-2\): \[ -2 \cdot (-x^2 + 3x - 6) = 2x^2 - 6x + 12 \]

Now, combine all terms: \[ (-x^4 + 3x^3 - 6x^2) + (x^3 - 3x^2 + 6x) + (2x^2 - 6x + 12) \]

Combine like terms:

  • \(-x^4\)
  • \((3x^3 + x^3) = 4x^3\)
  • \((-6x^2 - 3x^2 + 2x^2) = -7x^2\)
  • \((6x - 6x) = 0x\)
  • Constant term: \(12\)

Final expression in standard form: \[ -x^4 + 4x^3 - 7x^2 + 12 \]

Summary of Results:

  1. First Expression: \[ 2x^4 - 13x^3 + 13x^2 - 20x + 6 \]

  2. Second Expression: \[ -x^4 + 4x^3 - 7x^2 + 12 \]